ON THE HOMOTOPY THEORY ASSOCIATED TO CERTAIN FINITE
2-GROUPS OF 2-RANK TWO

By F.R. Cohen, J.R. Harper, and R. Levi

Appeared in Contemporary Mathematics 188 (1995) 65-79

Abstract

We study the homotopy theory associated to certain finite groups
\Gamma.  In particular, assume that \Gamma is 2-perfect.  Consider 
properties of the completion of  B\Gamma at the prime 2, B\Gamma^2, and its 
loop space where the 2-Sylow subgroup of \Gamma is \pi=SD_{2^n} the
semi-dihedral group of order 2^n.  The main result here is a comparison 
theorem which implies that \Omega(B\Gamma^2)$ can be ``resolved by finitely 
many spheres''. This comparison theorem holds in a mildly more general setting.
The homotopy groups of B\Gamma^2 are then shown to have some
possibly unusual properties.